Introduction to my May2002 Cookeville Clifford Algebra talk:
Cl(2N;C) = Cl(2;C) x ...(N times tensorproduct)... x Cl(2;C)
Cl(2;C) = M2(C) = 2x2 complex matrices
spinor representation = 1x2 complex columnspinors
Hyperfinite II1 vonNeumann Algebra factor is the completionof the union of all the tensor products
Cl(2;C) x ...(N times tensor product)... xCl(2;C)
By looking at the spinor representation, you seethat "the hyperfinite II1 factor is the smallest von Neumann algebracontaining the creation and annihilation operators on a fermionicFock space of countably infinite dimension."
In other words, Complex Clifford Periodicity leadsto the complex hyperfinite II1 factor which represents Dirac'selectron-positron fermionic Fock space.
Now, generalize this to get arepresentation of ALL the particles and fields ofphysics.
Use RealClifford Periodicity to construct aReal HyperfiniteII1 factor as the completion of the unionof all the tensor products
Cl(1,7;R) x ...(N times tensor product)... xCl(1,7;R)
where the Real Clifford Periodicity is
Cl(N,7N;R) = Cl(1,7;R) x ...(N times tensorproduct)... x Cl(1,7;R)
The components of the Real Hyperfinite II1 factorare each
[ my convention is (1,7) = (-+++++++) ]
Cl(1,7) is 2^8 = 16x16 =256-dimensional, and has graded structure
1 8 28 56 70 56 28 8 1
There are two mirror image half-spinors, each ofthe form of a real (1,7) column vector with octonionicstructure.
The 1 represents:
Second and third generation fermions come fromdimensional reduction of spacetime, so that
that reduces at lower energies to quaternionicstructures that are
There is a 28-dimensonal bivector representationthat corresponds to the gauge symmetry Lie algebraSpin(1,7)
that reduces at lower energiesto:
There is a 1-dimensional scalar representation forthe Higgsmechanism.
The above structures fit together to form aLagrangian
that reduces to a Lagrangian for Gravity plus theStandard Model.
Representations have geometric structure relatedto E6
E6 is an exceptional simple graded Lie algebra ofthe second kind:
E6 = g =g-2 + g-1 + g0 + g1 + g2
g0 = so(1,7) + R + iR
dim g-1 = 16
dim g-2 = 8
This gives real Shilovboundary geometry of S1xS7 for(1,7)-dimensional high-energy spacetime representation and for thefirst generation half-spinor fermion representations.
The geometry of the representation spaces, alongwith combinatorial structure of second and third generation fermions,allows calculation of relative force strengths and particlemasses:
?? Which is the TrueT-quark mass: 130 or 170 ??
E6 GLA structure is from Soji Kaneyuki's writing in Analysis andGeometry on Complex Homogeneous Domains, by Jacques Faraut, SojiKaneyuki, Adam Koranyi, Qi-keng Lu, and Guy Roos (Birkhauser 2000).